Energy and Temperature-Dependent Viscosity Analysis on Magnetized Eyring-Powell Fluid Oscillatory Flow in a Porous Channel

Abstract

In this research, we studied the impact of temperature dependent viscosity and thermal radiation on Eyring Powell fluid with porous channels. The dimensionless equations were solved using the perturbation technique using the Weissenberg number (ε ≪ 1) to obtain clear formulas for the velocity field. All of the solutions for the physical parameters of the Reynolds number (Re), magnetic parameter (M), Darcy parameter (Da) and Prandtl number (Pr) were discussed through their different values. As shown in the plots the two-dimensional and three-dimensional graphical results of the velocity profile against various pertinent parameters have been illustrated with physical reasons. The results revealed that the temperature distribution increases for higher Prandtl and thermal radiation values. Such findings are beneficial in the field of engineering sciences.

1. Introduction

In recent years, the mathematical and numerical simulation of non-Newtonian fluids has received particular attention from the research community. The analysis of such fluids has been intensively researched over the past decades due to their wide range of industrial applications in field such as petroleum engineering, energy engineering and food technology, etc. The most important non-Newtonian fluids include Eyring Powell fluid, nano fluid, Walters B fluid, and micropolar fluid, etc. [1,2]. Among the non-Newtonian fluids, Eyring Powell fluid has special rheological characteristics that make it preferable over other non-Newtonian fluids. High and low shear Eyring Powell fluid behaves similarly to Newtonian fluid. The Eyring Powell rights [3] are derived from the kinetic theory of gases, but not from the empirical relationship of the power law model. Extensive research on Eyring Powell fluid has been conducted due to its wide range of applications in businesses and industries. For example, Hina et al. [4] examined the thermophoric diffusion of Powell-Eyring fluid to a curved channel. In view of this, Longo et al. [5] studied non-Newtonian flow with gravity effects of the power law in an axially symmetric permeable medium. Similarly, Jalil, M., S. Asghar, and S. M. Imran [6] investigated the similarity solutions for Eyring Powell fluid flow passing through a moving plate. These researchers used a scaling group to transform the main equations into ordinary coupled differential equations, and Kellerbox techniques were used to solve the equations. Hayat et al. [7] discussed the behavior of Eyring Powell fluid in the presence of the parameters such as Dufour, Soret, Joule and thermal radiation.

It is worth discussing some investigations related to Eyring-Powell fluid here such as the study conducted by S. Nadeem and S. Sleem [8], who examined the unstable boundary layer flow of an Eyring Powell fluid with the combined effects of heat and mass transfer on a rotating cone. They used a series of homotopy solutions to obtain the solution for the mathematical model. Tanveer and Malik [9] studied the effect of slip and porous parameters in peristaltic flow under the influence of MHD. In the investigations of Hayat et al. [10], a moving surface with convective boundary conditions in a constant flow of Eyring Powell liquids was considered. In this study, an analytical method was applied to find the solution for the Blasius flow of Eyring Powell fluids with non-zero surface velocity conditions. A few more relevant investigations can be found in the references [11,12,13,14,15].

The applications of electrically conductive fluid currents encircled in the field of nanocomposite and metallurgy are introduced in this paragraph. Metallurgical requirements consist of continuous cooling belts or filaments such as hardening, disperse and sketching processes for copper wires. In all of these circumstances, the substances of the final product depend widely on the rate of cooling system. Therefore, to control the cooling and stretching rates, the magnetic field must be used intelligently. In recent decades, the problem of magneto hydro dynamic (MHD) flow in a rotating environment have received considerable interest due to their geographical, astrographs importance, and their uses in fluid technology. Various important problems have been discussed intensively, such as the preservations and worldly variations of the earth’s magnetic field due to themovement in the fluid in the center of the earth, the internal rotational speed of the sun, the structure of revolving magnetic stars, the global and solar doer problems, Ekman MHD pumps, turbomachines, rotary hydraulic generators and two-phase MHD rotary drum separators. The generator current in a closed circuit, is controlled by the action of magnetic forces. It has been observed that the effects of the Coriolis force are larger than those of viscosity and inertia forces in the hydro-magnetic equations of motion in a rotating environment. Several researchers have investigated the MHD with various kinds of fluid and geometries. Studies such as the one conducted by Abbas et.at [11] discussed the MHD peristaltic blood flow of nanofluid. They concluded that the magnetic field plays an important role in the blood pressure and temperature profile. The same authors [12] further studied that the entropy generation of MHD blood flow with the geometry of a peristaltic wave in complaint walls. They determined that magnetics field plays an important role in the velocity of blood particularly regarding its role in controlling bleeding during surgery. Bhatti et.al [13] examined the MHD and partial slip of Ree Ryring fluid with wall properties. They solved the simplified equation using an analytical method and concluded that the velocity distribution increases for larger slip parameter values while its opposite behavior was observed for various Hartmann parameter values. Wissam et.al [14] studied the impact of MHD on Jeffrey fluid with a porous channel saturated with temperature-dependent viscosity (TDV). Wissam et.al [15,16] studied the effect of heat transfer on MHD for Williamson fluid with a constant viscosity and variable viscosity through a porous medium. Other interesting investigations that are related MHD can be viewed in the references section of this paper [17,18,19].

Keeping the above discussions in view, much more attention is needed in the area of Eyring Powell fluid in the presence of MHD. Researchers, such as Dawar and Abdullah, et al. [17] have conducted a few investigations on Eyring Powell fluid through an oscillatory stretching sheet and focused on the comparison of analytical and numerical solutions. However, Eyring Powell fluid with temperature dependent properties is needed in industrial engineering because the Erying Powell fluid model is one of the non-Newtonian fluids and such fluid is widely used in ethylene glycol, kerosene, and water, which have demonstrated limited heat transport characteristics.

In the present study, a mathematical formulation of the problem in terms of continuity equation, momentum equations, and temperature equation has been developed. Further, dimensionless variables are introduced to simplify the governing equations and we obtained a nonlinear partial differential equation with boundary conditions. The solution of the nonlinear partial differential equation can be obtained by using the perturbation method, while the graphical interpretations of the velocity and temperature in two and three dimensions are discussed with physical reasons. The conclusions of the present analysis are discussed in the last section.

2. Mathematical Formulation

Consider the incompressible unstable flow for the magnetohydrodynamics of Jeffrey fluid with TDV in a saturated permeable channel with variable viscosity and at height $\left(L\right)$, as shown in Figure 1. It can be seen from the figure that we chose the Cartesian coordinates system $\left(x,y\right)$ as the velocity vector, in which $u \left(\mathrm{m}/\mathrm{s}\right)$ is the ($x$-component) of velocity, and $y \left(\mathrm{m}/\mathrm{s}\right)$ is perpendicular to the ($x$-axis). A uniform attractive field B₀ is forced along the y-axis.

The fundamental equation for Eyring-Powell fluid is given by:

$$S=-\overline{\mathrm{p}}I+\tau .$$

(1)

$$\overline{\tau }=\mu \left(T\right)\nabla \overline{V}+\frac{1}{B_1}{\sin \mathrm{h}}^{-1}\left(\frac{1}{A_1}\nabla \overline{V}\right).$$

(2)

where the parameters $\overline{\mathrm{p}}$, $I$, $\overline{\tau },$ $\mu \left(T\right)$ denote the pressure, unit tensor, extra stress tensor and variable shear rate viscosity while the parameter $\nabla \overline{V}$ is the velocity gradient.

The velocity field and the heat field of the present problem are:

$$V=\left[u\left(y,t\right),-v_0,0\right], T=T\left(y,t\right)$$

(3)

The continuity equation is given by:

$$\frac{\partial \overline{u}}{\partial \overline{x}}+\frac{\partial \overline{v}}{\partial \overline{y}}=0$$

(4)

The momentum equations are as follows:

In the $x$-direction:

$$\rho \frac{\partial \overline{u}}{\partial \overline{t}}-\rho v_0\frac{\partial \overline{u}}{\partial \overline{y}}=-\frac{\partial \overline{p}}{\partial \overline{x}}+\frac{\partial {\overline{\tau }}_{12}}{\partial \overline{y}}+\rho g\beta \left(T-T_0\right)-\sigma B_0^2\overline{u}-\frac{\mu \left(T\right)}{\rho K}\overline{u}$$

(5)

In the $y$-direction:

$$\mathsf{\rho }\left(\frac{\partial \overline{v}}{\partial \overline{t}}+\overline{u}\frac{\partial \overline{v}}{\partial \overline{x}}+\overline{v}\frac{\partial \overline{v}}{\partial \overline{y}}\right)=-\frac{\partial \overline{p}}{\partial \overline{y}}+\frac{\partial {\overline{\tau }}_{21}}{\partial \overline{x}}+\frac{\partial {\overline{\tau }}_{22}}{\partial \overline{y}}-\frac{\mu \left(T\right)}{\mathrm{k}}\overline{v}.$$

(6)

Temperature equation:

$$\rho \frac{\partial T}{\partial \overline{t}}-\frac{k}{C_p}\frac{{\partial }^{2}T}{\partial {\overline{y}}^2}-v_0\frac{\partial T}{\partial \overline{y}}=\frac{4b^2}{C_p}\left(T-T_0\right)$$

(7)

With associated boundary conditions

$$\begin{array}{c}\overline{u}=\frac{\sqrt{K}}{{\propto }_s}\frac{\partial \overline{u}}{\partial \overline{y}},T=T_0 \mathrm{at} \overline{y}=0\\ \overline{u}=0,T=T_1 \mathrm{at} \overline{y}=L\end{array}\}$$

(8)

The radioactive heat flux is given by:

$$\frac{\partial q}{\partial y}=4{\eta }^2\left(T_0-T\right).$$

(9)

where $\eta$ is the radiation absorption.

We can write the component of the extra stress tensor as follows:

$${\overline{\tau }}_{12}={\overline{\tau }}_{22}={\overline{\tau }}_{21}=0 {\overline{\tau }}_{12}=(\mu \left(\theta \right){\mu }_0+\frac{1}{B_1{A}_1})\frac{\partial \overline{u}}{\partial \overline{y}}-\frac{1}{6B_1{\left(A_1\right)}^3} {\left(\frac{\partial \overline{u}}{\partial \overline{y}}\right)}^3.$$

(10)

Dimensionless Formulation of the Mathematical Model

The non-dimensional conditions are as follows:

$$\begin{array}{c}x=\frac{\overline{x}}{\mathrm{h}}, y=\frac{\overline{y}}{\mathrm{h}}, u=\frac{\overline{u}}{\mathrm{U}}, \mathrm{p}=\frac{\overline{\mathrm{p}}\mathrm{h}}{\mu \mathrm{U}}, \mathrm{Pr}=\frac{{\mathsf{\rho }\mathrm{hUc}}_{\mathrm{p}}}{\mathrm{K}},S=\frac{v_{0}L{\mu }_0}{\rho } \\ \mathrm{t}=\frac{\overline{\mathrm{t}}\mathrm{U}}{\mathrm{h}}, {\tau }_{12}=\frac{\mathrm{h}}{{\mathsf{\mu }}_0\mathrm{U}}{\overline{\tau }}_{\overline{12}}, \mathrm{Gr}=\frac{\mathsf{\rho }g{\beta }_T{\mathrm{h}}^2\left(T_1-T_0\right)}{\mu \mathrm{U}} \\ \eta =\frac{\sqrt{K}}{{\propto }_s{L}^2},A=\frac{W}{6}{\left(\frac{\mathrm{U}}{\mathrm{h}{A}_1}\right)}^2, \mathrm{Re}=\frac{\mathsf{\rho }\mathrm{hU}}{\mu } , \mathrm{Da}=\frac{\mathrm{k}}{{\mathrm{h}}^2}, \theta =\frac{T-T_0}{T_1-T_0}\\ W=\frac{1}{{\mu }_0{B}_1{A}_1 }, \mu \left(\theta \right)=\frac{\mu \left(T\right)}{{\mu }_0}, N^2=\frac{4{\eta }^2{h}^2}{K},M^2=\frac{\sigma B_0^2{h}^2}{{\mu }_0} \end{array}\}$$

(11)

Using Equations (9)–(11) in Equations (4)–(8), we can creat the following non-dimensional equations:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.$$

(12)

$$\mathrm{Re}\frac{\partial u}{\partial t}=-\frac{d\mathrm{p}}{dx}+\frac{\partial {\tau }_{12}}{\partial y}+\mathrm{Gr}\theta -\left(M^2+\frac{\mu \left(\theta \right)}{\mathrm{Da}}\right)u.$$

(13)

$$\frac{\partial p}{\partial y}=0.$$

(14)

$$\frac{\partial \theta }{\partial t}-S\frac{\partial \theta }{\partial y}=\frac{1}{Pr}\frac{{\partial }^2\theta }{\partial y^2}+N^2\theta$$

(15)

$${\tau }_{12}=\left(\mu \left(\theta \right)+W\right)\frac{\partial u}{\partial y}-A{\left(\frac{\partial u}{\partial y}\right)}^3$$

(16)

With the boundary conditions

$$\begin{array}{c}u=\eta \frac{\partial y}{\partial y},\theta =0, at y=0\\ u=0 , \theta =1, at y=1\end{array}\}$$

(17)

After acquiring the simplifications, the resulting equation from placing Equation (16) into Equation (13), allow us to obtain the following equation:

$$\mathrm{Re}\frac{\partial u}{\partial t}=-\frac{d\mathrm{p}}{dx}+\mathrm{Gr}\theta -3A{\left(\frac{\partial u}{\partial y}\right)}^2\frac{{\partial }^{2}u}{\partial y^2}+\left(\mu \left(\theta \right)+W\right)\frac{{\partial }^{2}u}{\partial y^2}-\left(M^2+\frac{\mu \left(\theta \right)}{Da} \right)u.$$

(18)

3. Solution to the Problem

The exact solution of Equation (15) can be obtained as:

• Solution of the Heat Equations

To solve the temperature Equation (15), assume

$$\theta \left(y,t\right)={\theta }_f\left(y\right)e^{i\omega t}$$

(19)

Substituting Equation (19) into Equation (15), we obtained:

$$\frac{{\partial }^2{\theta }_f}{\partial y^2}+S·Pr\frac{\partial {\theta }_f}{\partial y}+\left(N^2-i\omega \right)Pr·{\theta }_f=0$$

(20)

The exact solution of Equations (19) and (20) is given by:

$${\theta }_f\left(y\right)=\frac{1}{-1+e^{\sqrt{{\mathsf{\Omega }}^2-4B}}}(e^{\frac{1}{2}\left(-\mathsf{\Omega }-\sqrt{{\mathsf{\Omega }}^2-4B}\right)y}\left(-e^{\frac{\mathsf{\Omega }}{2}+\frac{1}{2}\sqrt{{\mathsf{\Omega }}^2-4B}}\right)+e^{\frac{1}{2}\left(-\mathsf{\Omega }+\sqrt{{\mathsf{\Omega }}^2-4B}\right)y}\left(e^{\frac{\mathsf{\Omega }}{2}+\frac{1}{2}\sqrt{{\mathsf{\Omega }}^2-4B}}\right))$$

where $\mathsf{\Omega }=S.Pr, B=(N^2-i\omega )\times Pe$. Therefore:

$$\begin{gathered} \theta \left(y,t\right)=\frac{1}{-1+e^{\sqrt{{\mathsf{\Omega }}^2-4B}}} \\ (e^{\frac{1}{2}\left(-\mathsf{\Omega }-\sqrt{{\mathsf{\Omega }}^2-4B}\right)y}\left(-e^{\frac{\mathsf{\Omega }}{2}+\frac{1}{2}\sqrt{{\mathsf{\Omega }}^2-4B}}\right)+e^{\frac{1}{2}\left(-\mathsf{\Omega }+\sqrt{{\mathsf{\Omega }}^2-4B}\right)y}\left(e^{\frac{\mathsf{\Omega }}{2}+\frac{1}{2}\sqrt{{\mathsf{\Omega }}^2-4B}}\right))e^{i\omega t} \end{gathered}$$

(21)

ii.

Solution for Motion Equation

To solve the motion Equation (13) we assum,

$$\frac{d\mathrm{p}}{dx}=-\lambda e^{i\omega t}, u\left(y,t\right)=f\left(y\right)e^{i\omega t}.$$

(22)

where $\lambda , \omega$ denotes the real constant and the frequency of the oscillation.

The temperature in terms of Reynolds number and variation of viscosity are defined as

$$\mu \left(\theta \right)=e^{-\alpha \theta }.$$

(23)

By applying the Maclaurin series, we obtain:

$$\mu \left(\theta \right)=1-\alpha \theta \alpha <<1$$

(24)

At this stage, the viscosity is fixed at $\alpha =0$ by putting Equations (22) and (24) into Equation (17), we acquire:

$$\begin{gathered} Re\frac{\partial }{\partial t}\left(f\left(y\right)e^{i\omega t}\right)=\lambda e^{i\omega t}+\left(1-\alpha \theta \right)\frac{{\partial }^2}{\partial y^2}\left(f\left(y\right)e^{i\omega t}\right)+W\frac{{\partial }^2}{\partial y^2}\left(f\left(y\right)e^{i\omega t}\right)- \\ 3A\left(\frac{{\partial }^2}{\partial y^2}\left(f\left(y\right)e^{i\omega t}\right)\right){\left(\frac{\partial \left(f\left(y\right)e^{i\omega t}\right)}{\partial y}\right)}^2+Gr{\theta }_0\left(y\right)e^{i\omega t}-\left(M^2+\frac{\left(1-\alpha \theta \right)}{Da}\right)\left(f\left(y\right)e^{i\omega t}\right). \end{gathered}$$

(25)

Assuming the small value of $A,$ Equation (25) is a non-linear differential equation, so trying to find the exact solution is a complex task. To manage this situation, the perturbation method was used to find the problem’s solution, which was achieved as follows:

$$f=f_0+{\alpha }_1{f}_1+\mathrm{O}\left({\alpha }_1{}^2\right)$$

(26)

$$\begin{gathered} Re i\omega \left(f_0+{\alpha }_1{f}_1\right)=\lambda +\left(1-\alpha \theta \right)\frac{{\partial }^2}{\partial y^2}\left(f_0+{\alpha }_1{f}_1\right)+W\frac{{\partial }^2}{\partial y^2}\left(f_0+f_1\right)-3A\left(\frac{{\partial }^2}{\partial y^2}\left(f_0+{\alpha }_1{f}_1\right)\right){\left(\frac{\partial \left(f_0+A{f}_1\right)}{\partial y}\right)}^2 e^{2i\omega t}+ \\ Gr{\theta }_0-M^2\left(\left(f_0+{\alpha }_1{f}_1\right)\right)-\frac{\left(1-\alpha \theta \right)}{Da}\left(f_0+{\alpha }_1{f}_1\right). \end{gathered}$$

(27)

We exchanging Equation (26) into Equation (25), which is subject to the boundary conditions, and then by comparing the similar powers of $A$, we have:

• Zeros-order system ($A^0$)

$$Re i\omega f_0=\lambda +Gr{\theta }_0+\left(1-\alpha \theta \right)\frac{{\partial }^2{f}_0}{\partial y^2}+W\frac{{\partial }^2{f}_0}{\partial y^2}-\left(M^2+\frac{\left(1-\alpha \theta \right)}{Da}\right)f_0.$$

(28)

The associated boundary conditions are:

$$f_0=\eta \frac{\partial f_0}{\partial y} on y = 0 \mathrm{and} f_0 = 0 on y = 1$$

ii.

First-order system ($A^1$)

$$Re i\omega f_1=\left(1-\alpha \theta \right)\frac{{\partial }^2{f}_1}{\partial y^2}+W\frac{{\partial }^2{f}_1}{\partial y^2}-3e^{2i\omega t}\left(\frac{{\partial }^2{f}_0}{\partial y^2}\right){\left(\frac{\partial f_0}{\partial y}\right)}^2-\left(M^2+\frac{\left(1-\alpha \theta \right)}{Da}\right)f_{1 .}$$

(29)

The associated boundary conditions are:

$$f_1 = \eta \frac{\partial f_1}{\partial y} on y = 0 \mathrm{and} f_1 = 0 on y = 1$$

Through the expansion of the terms of $\left(A\right)$, we are able to derive Equations (28) and (29). We are able to interpret some of the natural explanations to this problem, which can be achieved by taking small values of $\alpha$ and taking a perturbation series with the parameters of $\alpha$. We substitute for $f_j$ (for $=0,1$ ) by expansion

$$f_j=f_{j0}+\alpha f_{j1}$$

(30)

We can then equate and compare the coefficients of similar power in $\alpha$, and the following set of equations can then be obtained.

3.1. Approximation of Solution for $f_0$

By substituting Equation (30) into Equation (28), we obtain

$$Re i\omega \left(f_{00}+\alpha f_{01}\right)=\lambda +Gr{\theta }_0+\left(1-\alpha \theta \right)\frac{{\partial }^2}{\partial y^2}\left(f_{00}+\alpha f_{01}\right)+W\frac{{\partial }^2}{\partial y^2}\left(f_{00}+\alpha f_{01}\right)-\left(M^2+\frac{\left(1-\alpha \theta \right)}{Da}\right)\left(f_{00}+\alpha f_{01}\right)$$

By equating the coefficients in the same way that we equated the powers of $\alpha$, we obtain:

• Zeros-Order System (${\alpha }^0$)

$$\frac{{\partial }^2{f}_{00}}{\partial y^2}-\left(\frac{i\omega Re+M^2+\frac{1}{Da}}{1+W}\right)f_{00}=-\left(\frac{\lambda +Gr{\theta }_0}{1+W}\right).$$

(31)

The associated boundary conditions are:

$$f_{00}=\eta \frac{\partial f_{00}}{\partial y} on y = 0 \mathrm{and} f_{00} = 0 on y = 1$$

ii.

First-Order System (${\alpha }^1$)

$$\frac{{\partial }^2{f}_{01}}{\partial y^2}-\left(\frac{i\omega Re+M^2+\frac{1}{Da}}{1+W}\right)f_{01}=\left(\frac{1}{1+W}\right)\left(\theta \frac{{\partial }^2{f}_{00}}{\partial y^2}-\left(\frac{\theta }{Da}\right)f_{00} \right)$$

(32)

The associated boundary conditions are:

$$f_{01}=\eta \frac{\partial f_{01}}{\partial y} on y = 0 \mathrm{and} f_{01} = 0 on y = 1$$

The perturbation solutions of Equations (31) and (32) and their boundary conditions are given:

$$f_0=f_{00}+\alpha f_{01}$$

3.2. Approximate Solution for $f_1$

By substituting $f_1$ given by the substituting Equation (30) into Equation (29), we obtain:

$$\begin{gathered} Re i\omega \left(f_{10}+\alpha f_{11}\right)=\left(1-\alpha \theta \right)\frac{{\partial }^2}{\partial y^2}\left(f_{10}+\alpha f_{11}\right)+W\frac{{\partial }^2}{\partial y^2}\left(f_{10}+\alpha f_{11}\right)-3e^{2i\omega t}\left(\frac{{\partial }^2}{\partial y^2}\left(f_{00}+\alpha f_{01}\right)\right){\left(\frac{\partial }{\partial y}\left(f_{00}+ \\ \alpha f_{01}\right)\right)}^2-\left(M^2+\frac{\left(1-\alpha \theta \right)}{Da}\right)\left(f_{10}+\alpha f_{11}\right) . \end{gathered}$$

Equating the coefficients in the same way as the powers in $\alpha$, were equated, we obtain:

• Zero-Order System (${\alpha }^0$)

$$\begin{gathered} \frac{{\partial }^2{f}_{10}}{\partial y^2}-\left(\frac{i\omega Re+M^2+\frac{1}{Da}}{1+W}\right)f_{10}=\frac{3}{1+W} e^{2i\omega t}\left(\frac{{\partial }^2{f}_{00}}{\partial y^2}\right){\left(\frac{\partial f_{00}}{\partial y}\right)}^2. \\ {f}_{10}=\eta \frac{\partial f_{00}}{\partial y} on y = 0 \mathrm{and} f_{10} = 0 on y = 1 \end{gathered}$$

(33)

ii.

First-Order System (${\alpha }^1$)

$$\frac{{\partial }^2{f}_{11}}{\partial y^2}-\left(\frac{i\omega Re+M^2+\frac{1}{Da}}{1+W}\right)f_{11}=\frac{1}{1+W}\left(\theta \frac{{\partial }^2{f}_{10}}{\partial y^2}+6e^{2i\omega t} \left(\frac{{\partial }^2{f}_{00}}{\partial y^2}\right)\left(\frac{\partial f_{00}}{\partial y}\right)\left(\frac{\partial f_{01}}{\partial y}\right)+3e^{2i\omega t}\left(\frac{{\partial }^2{f}_{01}}{\partial y^2}\right) {\left(\frac{\partial f_{00}}{\partial y}\right)}^2-\left(\frac{\theta }{Da}\right)f_{10}\right).$$

(34)

$$f_{11} = \eta \frac{\partial f_{00}}{\partial y}on y = 0 \mathrm{and} f_{11} = 0 on y = 1$$

The required solution of the perturbation of Equations (33) and (34), which are subject to the boundary conditions is as follows:

$$f_1=f_{10}+\alpha f_{11}$$

Finally, the solution up to second term for $f$ is provided by:

$$f=f_0+{\alpha }_1{f}_1$$

4. Results and Discussion

In this section, the magneto hydrodynamic oscillatory flow of Eyring Powell fluid with time-dependent viscosity in a permeable channel will be discussed. The perturbation method was employed to solve the governing equation, Equation (13), and the temperature equation, Equation (15). The solutions for the velocity profile and temperature distribution were performed using (Mathematical ver.11,) with the set of values$Re=1,\omega =\pi ,Pr=1,S=1,M=1,N=1,Gr=1,Da=0.8,\lambda =1,We=0.05,{\lambda }_1=0.3,ϵ=0.02,$ $\eta =0.1,t=0.5$.

As shown in Figure 2, the temperature profile increases for various cumulative oscillation frequencies $\omega$ during the canal decrease, $\theta$ is also clarified. The variations of $S$ on $\theta$ are shown in Figure 3. From this figure, it is revealed that the fluid temperature is linearly dispersed in the channel. Conversely, the fluid temperature in the channel increases with the increase of the injection on the heated bowl. The linearity inspected at $S=0$ provides an approach to concave circulation. By increasing $S$, the concavity is an important factor due to the pathway of the flow of temperature which is absorbed from the animated superficial to the cold superficial each time. Figure 4 displays that $\theta$ increases with the increase in the value of $N$ because the temperature is communicated from the animated wall near the fluid.

Figure 5 shows that velocity distribution increases with increasing of the parameter $Da$. Figure 6 illustrates the effect of $Gr$ on the velocity field. It demonstrates that after increasing $Gr$, the velocity field $u$ increases. This phenomenon occurs when buoyancy forces are high in association with viscous forces. The effects of the parameter $\lambda$ on velocity profile can be seen from Figure 7. By increasing $\lambda$, velocity distribution $u$ increases. While Figure 8 shows the influence $M$ on the velocity field function $u$. By increasing $M$ the velocity distribution is decreased. It should be noted that the change in velocity distribution is associated with external magnetic field and the increase in velocity is accredited with increase in Lorentz force.

Figure 9 illustrates that by increasing $N$, the velocity distribution $u$ increases. The reason behind this behavior is that the fluid becomes thinner due to higher values in the radiation parameter. Figure 10 displays that increasing $Pr$ leads to an increase in the velocity distribution $u$. For larger Prandtl number values the, diffusivity rise and the heat diffuses very quickly. Therefore, velocity is affected and increase. Figure 11 illustrates the fact that the velocity distribution $u$ is decreases when$Re$ increases. The solution obtained from Equation (13) clearly shows that $Re$ is inversely proportional to the velocity profile due to velocity decreasing with $Re$. Figure 12 shows that increasing $S$ results in an increase in the velocity field. Figure 13 shows that velocity $\theta$ decreases when $\omega$ increases. All of the graphs depicting the velocity profile have been plotted in 3D to achieve a clear visibility and better interpretations and analysis.

5. Conclusions

The oscillatory magneto hydrodynamic flow problem of Eyring Powell fluid with temperature-dependent viscosity (TDV) in a saturated porous channel has been comprehensively discussed in this paper. The solutions of the velocity profile and temperature distribution were satisfactorily obtained using the perturbation method. Different sets of values have been employed to find the solution of the mathematical model. Finally, it was observed that:

• Increasing the radiation parameter and Prandtl number increases the velocity profile.
• Increase in $M$ and $Re$ lead to a decreasing pattern in the velocity field due to the existence of the Lorentz force in the flow, thereby generating resistance.
• Higher radiation parameter and injection parameter values cause increases in the temperature profile.
• Temperature decreases when the $\omega$ increases.